class: center, middle, inverse, title-slide .title[ # Class 5a: Multiple Linear Regression ] .author[ ### Business Forecasting ] --- <style type="text/css"> .remark-slide-content { font-size: 20px; } </style> --- ## Roadmap ### This set of classes - What is a multiple linear regression --- ### Motivation - Suppose that you are managing a hospital - You want to predict how long a patient will stay at the urgent care - You need this information to see how many rescources you need (doctors, beds, etc) -- - You collect the data on - The Duration of the visit - Characteristics of the patient - What kind of problem the patient came with - What type of bed they got - How many other patients there are currently at urgent care -- - If we know these values, can we predict how long patient will stay? --- ### Data <div class="datatables html-widget html-fill-item" id="htmlwidget-399af8abc07320838651" style="width:100%;height:auto;"></div> <script type="application/json" 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ESPECIFICADO","MÉDICA","MÉDICA","MÉDICA"]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>ID<\/th>\n <th>Duration<\/th>\n <th>Occupancy<\/th>\n <th>SEXO<\/th>\n <th>EDAD<\/th>\n <th>TIPOCAMA<\/th>\n <th>MOTATE<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":10,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[0,1,2,4]},{"name":"ID","targets":0},{"name":"Duration","targets":1},{"name":"Occupancy","targets":2},{"name":"SEXO","targets":3},{"name":"EDAD","targets":4},{"name":"TIPOCAMA","targets":5},{"name":"MOTATE","targets":6}],"order":[],"autoWidth":false,"orderClasses":false},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> --- ### Multiple linear regression - Multiple linear regression is similar to single linear regression -- - But you can use more than just one x (predictor) -- - We model y as a linear function of multiple variables -- - Since you can use many explanatory variables (Xs) you can usually get much better predictions --- ### Multiple linear regression - Back to hospital example: Suppose that the outcome `\(y_i\)` (duration) is a linear function of `\(x_1\)` (Occupancy) and `\(x_2\)` (age) `$$y_i=\beta_0+\beta_1x_{i1}+\beta_2x_{i2}+u_i$$` - `\(\beta_0\)` represents the average value of `\(y_i\)` when `\(x_1\)` and `\(x_2\)` are 0. - `\(\beta_1\)` represents the change in `\(y_i\)` while changing `\(x_1\)` by one unit and keeping `\(x_2\)` constant - `\(\beta_2\)` represents the change in `\(y_i\)` while changing `\(x_2\)` by one unit and keeping `\(x_1\)` constant -- Once I estimated the parameters, I can predict how long a patient will stay. Consider a patient who is 20 years old (age) and there are 5 other people in the urgent care at the same time (occupancy), then predicted stay is: `$$\hat{y}_i=\hat{\beta}_0+\hat{\beta}_1*5+\hat{\beta}_2*10$$` --- ### Example 1 What can predict hourly wage in the US? -- ``` ## ## Call: ## lm(formula = wage ~ exper + looks + female + married + educ, ## data = beauty) ## ## Residuals: ## Min 1Q Median 3Q Max ## -7.520 -2.190 -0.599 1.149 72.998 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -1.66350 0.86433 -1.925 0.0545 . ## exper 0.08280 0.01065 7.778 1.53e-14 *** ## looks 0.39694 0.17618 2.253 0.0244 * ## female -2.37281 0.26662 -8.899 < 2e-16 *** ## married 0.69044 0.27502 2.511 0.0122 * ## educ 0.44112 0.04623 9.542 < 2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 4.192 on 1254 degrees of freedom ## Multiple R-squared: 0.1944, Adjusted R-squared: 0.1912 ## F-statistic: 60.52 on 5 and 1254 DF, p-value: < 2.2e-16 ``` --- ### Example 2 What can predict prices of houses in the US? -- ``` ## ## Call: ## lm(formula = price ~ rooms + baths + age + land, data = hprice3) ## ## Residuals: ## Min 1Q Median 3Q Max ## -92853 -24120 -2654 16466 166082 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -8.942e+03 1.403e+04 -0.637 0.52449 ## rooms 6.647e+03 2.646e+03 2.512 0.01249 * ## baths 2.674e+04 3.337e+03 8.013 2.18e-14 *** ## age -1.975e+02 6.246e+01 -3.163 0.00172 ** ## land 5.695e-02 4.833e-02 1.178 0.23952 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 33130 on 316 degrees of freedom ## Multiple R-squared: 0.42, Adjusted R-squared: 0.4126 ## F-statistic: 57.2 on 4 and 316 DF, p-value: < 2.2e-16 ``` --- ### Example 3 What can predict whether you liked someone romantically? -- ``` ## ## Call: ## lm(formula = Like ~ Attractive + Intelligent + Fun + Ambitious + ## SharedInterests, data = SpeedDating2) ## ## Residuals: ## Min 1Q Median 3Q Max ## -4.2155 -0.6209 0.0768 0.6328 4.0277 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 0.30415 0.29653 1.026 0.30556 ## Attractive 0.39021 0.03440 11.345 < 2e-16 *** ## Intelligent 0.16475 0.04565 3.609 0.00034 *** ## Fun 0.23172 0.03686 6.286 7.45e-10 *** ## Ambitious -0.02309 0.03612 -0.639 0.52305 ## SharedInterests 0.18108 0.02832 6.393 3.92e-10 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 1.113 on 470 degrees of freedom ## (76 observations deleted due to missingness) ## Multiple R-squared: 0.6176, Adjusted R-squared: 0.6135 ## F-statistic: 151.8 on 5 and 470 DF, p-value: < 2.2e-16 ``` --- ### Multiple linear regression - With one regressor, relationship was visualized with a line, what about now? -- 100 observations simulated from a regression line: `$$y_i=5+2x_{i1}+1x_{i2}+u_i$$` <div id="htmlwidget-30cd82b6743fb072eb3d" style="width:80%;height:400px;" class="plotly html-widget"></div> <script type="application/json" 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12.5366542090647"],"hoverinfo":"text","mode":"markers","type":"surface","opacity":0.59999999999999998,"frame":null}],"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.20000000000000001,"selected":{"opacity":1},"debounce":0},"shinyEvents":["plotly_hover","plotly_click","plotly_selected","plotly_relayout","plotly_brushed","plotly_brushing","plotly_clickannotation","plotly_doubleclick","plotly_deselect","plotly_afterplot","plotly_sunburstclick"],"base_url":"https://plot.ly"},"evals":[],"jsHooks":[]}</script> --- ### Multiple linear regression 100 observations simulated from a regression line: `$$y_i=5+2\underbrace{x_{i}}_{x_{i1}}-1\underbrace{x_i}_{x_{i2}}^2+u_i$$` <img src="data:image/png;base64,#C_5_slides_a_files/figure-html/unnamed-chunk-6-1.png" width="100%" /> --- ### Multiple linear regression Suppose that: `$$x_1 = \begin{cases} 1 & \text{if female} \\ 0 & \text{if male} \end{cases}$$` 100 observations simulated from a regression line: `$$y_i=5+2x_{i1}-1x_{i2}+u_i$$` <img src="data:image/png;base64,#C_5_slides_a_files/figure-html/unnamed-chunk-7-1.png" width="100%" /> --- ### Multiple linear regression Now imagine a regression with k variables: `$$y_i=\beta_0+\beta_1x_{i1}+\beta_2x_{i2}+...+\beta_kx_{ik}+u_i$$` - Maybe you are trying to predict customer spending based on what they looked at and `\(x_{ij}\)` represent how long customer `\(i\)` looked at item `\(j\)` -- - Maybe you are trying to predict sales in a store `\(i\)`, and `\(x_{ij}\)` represent prices of the products, their competitors' products, how many people live around and how rich are they etc... -- - We can no longer visualize it (because we can't visualize more than 3 dimensions) --- ### Multiple linear regression We can also write the regression in the vector form: `$$y_i=\beta_0+\beta_1x_{i1}+\beta_2x_{i2}+...+\beta_k,x_{ik}+u_i$$` In vector form is: `$$\mathbf{y}=\mathbf{X\beta}+\mathbf{u}$$` <div class="math"> \[ \underbrace{\begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \\ \end{bmatrix}}_{\substack{\mathbf{y} \\ n \times 1}} = \underbrace{\begin{bmatrix} 1 & x_{11} & x_{12} & ... & x_{1k} \\ 1 & x_{21} & x_{22} & ... & x_{2k} \\ \vdots & \vdots & \vdots & ....& \vdots \\ 1 & x_{n1} & x_{n2} & ... & x_{nk} & \end{bmatrix}}_{\substack{\mathbf{X} \\ n \times (k+1)}} \underbrace{\begin{bmatrix} \beta_0 \\ \beta_1 \\ \vdots \\ \beta_k \\ \end{bmatrix}}_{\substack{\mathbf{\beta} \\ (k+1) \times 1}} + \underbrace{\begin{bmatrix} u_1 \\ u_2 \\ \vdots \\ u_n \\ \end{bmatrix}}_{\substack{\mathbf{u} \\ n \times 1}} \] </div> --- ### Full Rank Important Assumption: **X is full rank** - Has same rank as the number of parameters: `\(p=k+1\)` - Also known as: no perfect multicolinearity -- - .blue[Technically]: columns of X should be linearly independent -- - .blue[Intuitively]: none of the variables are perfectly correlated. If they are perfectly correlated, then we don't need one of the columns because we can perfectly predict one column with information from another column. - Suppose that one column is income in USD, and the second one is income measured in Pesos. They are perfectly correlated. Once we know income in USD, income in Pesos does not bring any additional information. We would not be able to estimate the effect of both income in USD and income in Pesos at the same time. <div class="math"> \[ \begin{array}{cc} \text{Full Rank Matrix:} & \text{Matrix Not of Full Rank:} \\ \left[\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right] & \left[\begin{array}{ccc} 1 & 2 & 4 \\ 4 & 5 & 10 \\ 7 & 8 & 16 \end{array}\right] \end{array} \] </div> --- ### Multiple linear regression **Goal:** - Estimate the vector of parameters `\(\mathbf{\beta}\)` **Procedure** - Find <div class="math"> \[ \mathbf{b}=\begin{bmatrix} b_0 \\ b_1 \\ \vdots \\ b_k \\ \end{bmatrix} \] </div> - Which minimizes the squared errors in the problem: `$$y_i=b_0+b_1x_{i1}+b_2x_{i2}+...+b_kx_{ik}+e_i$$` - That is minimize `$$SSE=\sum_ie_i^2=\sum_i(y_i-\hat{y}_i)^2=\mathbf{e}'\mathbf{e}=(\mathbf{y-\hat{y}})'(\mathbf{y-\hat{y}})=(\mathbf{y-Xb})'(\mathbf{y-Xb})$$` --- ### Multiple linear regression - We can do it with scalars `$$\begin{align*} \frac{\partial SSE}{\partial \hat{\beta}_0} & = -2\sum_{i=1}^{n} \left( y_i - (\hat{\beta}_0 + \hat{\beta}_1 x_{i1}+...+\hat{\beta}_k x_{ik})\right)=0 \\ \frac{\partial SSE}{\partial \hat{\beta}_1} & = -2\sum_{i=1}^{n} x_{i1} \left( y_i - (\hat{\beta}_0 + \hat{\beta}_1 x_{i1}+...+\hat{\beta}_k x_{ik})\right)=0 \\ \vdots \\ \frac{\partial SSE}{\partial \hat{\beta}_k} & = -2\sum_{i=1}^{n} x_{ik} \left( y_i - (\hat{\beta}_0 + \hat{\beta}_1 x_{i1}+...+\hat{\beta}_k x_{ik})\right)=0 \\ \end{align*}$$` -- - We have `\(k+1\)` equations with `\(k+1\)` unknowns. --- ### Multiple linear regression - Or we can do it with vectors -- - First rewrite the sum of squares: `$$\small SSE(b)=(\mathbf{y-Xb})'(\mathbf{y-Xb})=\mathbf{y'}\mathbf{y-2b'X'y}+\mathbf{b'X'Xb}$$` - Short review of matrix derivatives (link to full review of matrix algebra in canvas): 1. Constants drop out: `$$\small \frac{\partial}{\partial \mathbf b}(\mathbf v) = 0$$` 2. Linear functions: `$$\frac{\partial}{\partial \mathbf b}(\mathbf c' \mathbf b) = \mathbf c, \qquad \frac{\partial}{\partial \mathbf b}(\mathbf b' \mathbf c) = \mathbf c$$` 3. Quadratic forms $$ \small \frac{\partial}{\partial \mathbf b}(\mathbf b' A \mathbf b) = (A + A')\mathbf b $$ --- ### Multiple linear regression `$$SSE(b)=(\mathbf{y-Xb})'(\mathbf{y-Xb})=\mathbf{y'}\mathbf{y-2b'X'y}+\mathbf{b'X'Xb}$$` - Let's minimize it with respect to `\(\mathbf{b}\)` `$$\frac{\partial}{\partial \mathbf{b}}(\mathbf{y'}\mathbf{y-2b'X'y}+\mathbf{b'X'Xb})=\mathbf{-2X'y}+\mathbf{2X'Xb}$$` -- - `\(\hat{\beta}\)` is the solution of such minimization (our OLS estimator) `$$\begin{align*} \mathbf{-2X'y}+\mathbf{2X'X\hat{\beta}}&=0 \\ \mathbf{X'X\hat{\beta}} & =\mathbf{X'y} \\ \mathbf{\hat{\beta}} & =\mathbf{(X'X)^{-1}X'y} \end{align*}$$` --- ### Multiple linear regression Looking more closely at the **first order condition**: <div class="math"> \[ \underbrace{\begin{bmatrix} n & \sum_{i=1}^{n}x_{i1} & \ldots & \sum_{i=1}^{n} x_{ik} \\ \sum_{i=1}^{n} x_{i1} & \sum_{i=1}^{n} x_{i1}^2 & \ldots & \sum_{i=1}^{n} x_{i1}x_{ik} \\ \vdots & \vdots & \ddots & \vdots \\ \sum_{i=1}^{n} x_{ik}& \sum_{i=1}^{n} x_{ik}x_{i1} & \ldots & \sum_{i=1}^{n} x_{ik}^2\end{bmatrix}}_{\mathbf{X'X}} \underbrace{\begin{bmatrix} \hat{\beta}_0 \\ \hat{\beta}_1 \\ \vdots \\ \hat{\beta}_k\end{bmatrix}}_{\hat{\beta}} = \underbrace{\begin{bmatrix} \sum_{i=1}^{n}y_i \\ \sum_{i=1}^{n}x_{i1}y_i \\ \vdots \\ \sum_{i=1}^{n}x_{ik}y_i\end{bmatrix}}_{\mathbf{X'y}} \] </div> Looking more closely and it's **solution**: <div class="math"> \[\underbrace{\begin{bmatrix} \hat{\beta}_0 \\ \hat{\beta}_1 \\ \vdots \\ \hat{\beta}_k\end{bmatrix}}_{\hat{\beta}} = \underbrace{\begin{bmatrix} n & \sum_{i=1}^{n}x_{i1} & \ldots & \sum_{i=1}^{n} x_{ik} \\ \sum_{i=1}^{n} x_{i1} & \sum_{i=1}^{n} x_{i1}^2 & \ldots & \sum_{i=1}^{n} x_{i1}x_{ik} \\ \vdots & \vdots & \ddots & \vdots \\ \sum_{i=1}^{n} x_{ik}& \sum_{i=1}^{n} x_{ik}x_{i1} & \ldots & \sum_{i=1}^{n} x_{ik}^2\end{bmatrix}^{-1}}_{\mathbf{(X'X)}^{-1}} \underbrace{\begin{bmatrix} \sum_{i=1}^{n}y_i \\ \sum_{i=1}^{n}x_{i1}y_i \\ \vdots \\ \sum_{i=1}^{n}x_{ik}y_i\end{bmatrix}}_{\mathbf{X'y}} \] </div> --- ### Predictions To make predictions based on the estimated regressors we use: `$$\hat{y}_i=\hat{\beta}_0+\hat{\beta}_1x_{i1}+\hat{\beta}_2x_{i2}+...+\hat{\beta}_kx_{ik}$$` Or in the vector form: `$$\mathbf{\hat{y}}=\mathbf{X\hat{\beta}}=\mathbf{X\mathbf{(X'X)}^{-1}\mathbf{X'y}}=\mathbf{Hy}$$` Where `\(\mathbf{H}=\mathbf{X(X'X)}^{-1}\mathbf{X'}\)` is called a hat matrix. --- ### Residuals To get residuals, we calculate: `$$e_i=y_i-\hat{y}_i=y_i-\hat{\beta}_0+\hat{\beta}_1x_{i1}+\hat{\beta}_2x_{i2}+...+\hat{\beta}_kx_{ik}$$` Or in the vector form: `$$\mathbf{e}=\mathbf{y-\hat{y}}=y-\mathbf{X\hat{\beta}}=\mathbf{y}-\mathbf{X\mathbf{(X'X)}^{-1}\mathbf{X'y}}=\mathbf{(I-H)y}$$` --- ### Summary - We are trying to find `\(\beta\)`s which minimize the prediction error - It turns out that we get minimal errors when we set `\(\beta\)` to be: `\(\hat{\beta}=(X'X)^{-1}X'y\)` - If we have just one regressors, we get back the same formular as we already derived --- #### Example with numbers What's the impact of hours studied and hours slept on the exam score? `$$\small \begin{align*} \text{Dataset:} \\ &\begin{array}{|c|c|c|c|} \hline \text{Student} & \text{Hours Studied (}x_1\text{)} & \text{Hours Slept (}x_2\text{)} & \text{Exam Score (}y\text{)} \\ \hline 1 & 3 & 8 & 80 \\ 2 & 4 & 7 & 85 \\ 3 & 6 & 6 & 92 \\ 4 & 5 & 7 & 88 \\ \hline \end{array} \\ \text{X matrix}: \\ & X = \begin{bmatrix} 1 & 3 & 8 \\ 1 & 4 & 7 \\ 1 & 6 & 6 \\ 1 & 5 & 7 \\ \end{bmatrix} \\ \text{Response Vector } (y): \\ & y = \begin{bmatrix} 80 \\ 85 \\ 92 \\ 88 \\ \end{bmatrix} \end{align*}$$` --- ## What these matrices mean and do?! - Now we have more regressors, and we want to know SEPARATE impact of each. - What would happen if we change just that one thing (hours of sleep) and keep everything else constant (hours studied)? -- - But how do we keep other things constant? Hours slept and hours studied could be correlated -- - So when we compare people with different hours of sleep, there is risk we also compare people with different hours studied -- - How do we know that the change we see is coming from hours of sleep, not hours of studied? -- - Cross terms in the matrix discount this correlation - Intuitively, this makes variables "uncorrelated" and allows to get their own impacts. --- #### Example with numbers Multiply `\(X'\)` by `\(X\)`: `$$X'X = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 3 & 4 & 6 & 5 \\ 8 & 7 & 6 & 7 \\ \end{bmatrix} \begin{bmatrix} 1 & 3 & 8 \\ 1 & 4 & 7 \\ 1 & 6 & 6 \\ 1 & 5 & 7 \\ \end{bmatrix}= \begin{bmatrix} 4 & 18 & 28 \\ 18 & 86 & 123 \\ 28 & 123 & 198 \\ \end{bmatrix}$$` Find the inverse `\((X'X)^{-1}\)` `$$(X'X)^{-1} = \begin{bmatrix} 474.75 & -30 & -48.5 \\ -30 & 2 & 3 \\ -48.5 & 3 & 5 \\ \end{bmatrix}$$` --- #### Example with numbers Next let's find `\(X'y\)` `$$X'y= \begin{bmatrix} 1 & 1 & 1 & 1 \\ 3 & 4 & 6 & 5 \\ 8 & 7 & 6 & 7 \\ \end{bmatrix} \begin{bmatrix} 80 \\ 85 \\ 92 \\ 88 \\ \end{bmatrix}= \begin{bmatrix} 345 \\ 1572 \\ 2403 \\ \end{bmatrix}$$` So, our coefficients are: `$$\beta=\begin{bmatrix} \hat{\beta}_0 \\ \hat{\beta}_1 \\ \hat{\beta}_2 \\ \end{bmatrix} = \underbrace{\begin{bmatrix} 474.75 & -30 & -48.5 \\ -30 & 2 & 3 \\ -48.5 & 3 & 5 \\ \end{bmatrix}}_{(X'X)^{-1}} \underbrace{\begin{bmatrix} 345 \\ 1572 \\ 2403 \\ \end{bmatrix}}_{X'y}= \begin{bmatrix} 83.25 \\ 3 \\ -1.5 \\ \end{bmatrix}$$` -- **Interpretation** - Score with 0 hours of sleep and 0 of studying is 83.25 - 1 more hour of studying (without changing sleep hours) increases score by 3 - 1 more hour of sleep (without changing study hours) decreases score by 1.5 --- #### Example with numbers We can find predicted values: `$$\hat{y}=X\hat{\beta}= \begin{bmatrix} 1 & 3 & 8 \\ 1 & 4 & 7 \\ 1 & 6 & 6 \\ 1 & 5 & 7 \\ \end{bmatrix}\begin{bmatrix} 83.25 \\ 3 \\ -1.5 \\ \end{bmatrix}= \begin{bmatrix} 80.25 \\ 84.75 \\ 92.25 \\ 87.75 \\ \end{bmatrix}$$` And the residuals: `$$e=y-\hat{y}=y-X\hat{\beta}= \begin{bmatrix} 80 \\ 85 \\ 92 \\ 88 \\ \end{bmatrix}- \begin{bmatrix} 80.25 \\ 84.75 \\ 92.25 \\ 87.75 \end{bmatrix}= \begin{bmatrix} -0.25 \\ 0.25 \\ -0.25 \\ 0.25 \end{bmatrix}$$` --- #### Example from data: ``` r # Fit a linear regression model lm_model <- lm(Duration ~ Occupancy+EDAD, data = Sample_urg) # Display the summary of the linear regression model summary(lm_model) ``` ``` ## ## Call: ## lm(formula = Duration ~ Occupancy + EDAD, data = Sample_urg) ## ## Residuals: ## Min 1Q Median 3Q Max ## -773.65 -26.61 -17.27 -0.57 1252.75 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 23.23422 2.48416 9.353 < 2e-16 *** ## Occupancy 3.70354 0.10090 36.705 < 2e-16 *** ## EDAD 0.20626 0.06747 3.057 0.00225 ** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 98.99 on 4995 degrees of freedom ## Multiple R-squared: 0.2169, Adjusted R-squared: 0.2166 ## F-statistic: 691.8 on 2 and 4995 DF, p-value: < 2.2e-16 ``` --- ### Practice Predict how long a patient will stay if there 10 other patients in the hospital and the patient is 50 years old. --- ### Correlations vs Coefficients Note, that `\(x_1\)` and `\(x_2\)` can both have positive correlation with `\(y_i\)`, but different coefficients! - Suppose `\(x_1\)` is study hours, `\(x_2\)` is coffee cups drunk by a student, and `\(y\)` is student's score on the exam. <img src="data:image/png;base64,#C_5_slides_a_files/figure-html/unnamed-chunk-9-1.png" width="1000%" /> --- ### Correlations vs Coefficients ``` ## ## Call: ## lm(formula = y ~ x1 + x2, data = data) ## ## Residuals: ## Min 1Q Median 3Q Max ## -2.779 -1.422 -0.418 1.096 6.305 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 3.13966 0.38033 8.255 7.68e-13 *** ## x1 2.06132 0.11686 17.640 < 2e-16 *** ## x2 -0.08510 0.09798 -0.868 0.387 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 1.88 on 97 degrees of freedom ## Multiple R-squared: 0.9018, Adjusted R-squared: 0.8997 ## F-statistic: 445.2 on 2 and 97 DF, p-value: < 2.2e-16 ``` -- - Why coffee has 0 impact? -- - Because it only helps to study longer, but comparing students who study the same amount, drinking more coffee is not better. --- ### More realistic example from the data: Let’s explore a simple linear regression using U.S. survey data. We’ll regress hourly wage (in cents) on years of education. ``` r # Fit a linear regression model lm_model <- lm(wage ~ educ, data = card) # Display the summary of the linear regression model summary(lm_model) ``` ``` ## ## Call: ## lm(formula = wage ~ educ, data = card) ## ## Residuals: ## Min 1Q Median 3Q Max ## -576.09 -173.36 -34.12 127.82 1686.25 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 183.949 23.104 7.962 2.38e-15 *** ## educ 29.655 1.708 17.368 < 2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 250.7 on 3008 degrees of freedom ## Multiple R-squared: 0.09114, Adjusted R-squared: 0.09084 ## F-statistic: 301.6 on 1 and 3008 DF, p-value: < 2.2e-16 ``` --- ### More realistic example from the data: Now let’s include IQ as an additional predictor: The coefficient on education decreases. Why? ``` r # Fit a linear regression model # Fit a linear regression model lm_model <- lm(wage ~ educ+IQ, data = card) # Display the summary of the linear regression model summary(lm_model) ``` ``` ## ## Call: ## lm(formula = wage ~ educ + IQ, data = card) ## ## Residuals: ## Min 1Q Median 3Q Max ## -589.77 -171.71 -24.22 126.61 1672.88 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 149.1893 41.8965 3.561 0.000378 *** ## educ 20.7319 2.8829 7.191 8.94e-13 *** ## IQ 1.7253 0.4251 4.059 5.11e-05 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 255.9 on 2058 degrees of freedom ## (949 observations deleted due to missingness) ## Multiple R-squared: 0.06048, Adjusted R-squared: 0.05956 ## F-statistic: 66.24 on 2 and 2058 DF, p-value: < 2.2e-16 ``` --- ### Why Does the Coefficient Change? - In the first model, education captures both its own effect and part of IQ’s effect on wage, because education and IQ are positively correlated. - We're comparing people with different education levels, who also tend to differ in IQ. - In the second model, we control for IQ explicitly. - Now, we are comparing people with different education levels, but with the same IQ level (holding IQ fixed). - Result: The education coefficient shrinks because it no longer absorbs IQ’s influence. --- ### OLS Properties - As usual, we asked whether it's unbiased and what is its variance. -- - **Unbiased**: <div class="math"> \[\small \begin{align*} E(\hat{\beta}) & =E(\mathbf{(X'X)^{-1}X'y})=E(\mathbf{(X'X)^{-1}X'(X\beta+u)})) \\ & = E(\mathbf{(X'X)^{-1}X'(X\beta+u)}))=E(\mathbf{(X'X)^{-1}X'X\beta})+E(\mathbf{(X'X)^{-1}X'u}) \\ & = \beta+\mathbf{(X'X)^{-1}X'E(u)}=\beta+0+\beta \end{align*} \] </div> Where `\(\small E(\mathbf{(X'X)^{-1}X'u})\)` if `\(\small E(u)=0\)` (our usual assumption). -- - **Variance** `$$\small Var(\hat{\beta})=Cov(\hat{\beta})=\underbrace{\begin{bmatrix} var(\hat{\beta_0}) & cov(\hat{\beta_0}, \hat{\beta_1}) & ... & cov(\hat{\beta_0}, \hat{\beta_k}) \\ cov(\hat{\beta_1}, \hat{\beta_0}) & var(\hat{\beta_1}) & ... & cov(\hat{\beta_1}, \hat{\beta_k}) \\ \vdots & \vdots & \vdots & \vdots \\ cov(\hat{\beta_k}, \hat{\beta_0}) & cov(\hat{\beta_k}, \hat{\beta_1}) & ... & var(\hat{\beta_k}) & \end{bmatrix}}_{\substack{ \\ (k+1) \times (k+1)}}$$` - So it's a matrix with variance of single parameters on the diagonal and covariances off the diagonal. --- ## Example  --- ### Variance First, note that: `$$\hat{\beta} = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'(\mathbf{X}\beta + \mathbf{u}) = \beta + (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{u}$$` Let's use this `$$\small \begin{align*} var(\hat{\beta}) & = \mathbb{E}[(\hat{\beta} - \mathbb{E}[\hat{\beta}]) (\hat{\beta} - \mathbb{E}[\hat{\beta}])'] \\ & = \mathbb{E}[(X'X)^{-1}X'\mathbf{u} ((X'X)^{-1}X'\mathbf{u})']= (X'X)^{-1}X'\mathbb{E}[\mathbf{u}\mathbf{u}']X(X'X)^{-1} \\ & = (X'X)^{-1}X'(I\sigma^2)X(X'X)^{-1} =\sigma^2(X'X)^{-1} \end{align*}$$` -- - Where two last inequalies come from? - From `\(\sigma^2\)` being constant and errors having zero covariance. -- So `$$var(\hat{\beta}_k)=\sigma^2(X'X)^{-1}_{k+1,k+1}$$` where `\((X'X)^{-1}_{k+1,k+1}\)` is element in `\(k+1\)` row and `\(k+1\)` column of `\((X'X)^{-1}\)` matrix. First one is intercept! --- ### Example: `$$\text{cov}(\hat{\beta})= \begin{bmatrix} 0.07784 & -0.00808 & -0.00738 \\ -0.00808 & 0.0175 & -0.00014 \\ -0.00738 & -0.00014 & 0.00164 \end{bmatrix}=$$` And the residuals: `$$\text{cov}(\hat{\beta})=\underbrace{0.02389}_{\sigma^2} \times \underbrace{\begin{bmatrix} 3.2583 & -0.3382 & -0.3089 \\ -0.3382 & 0.7325 & -0.0059 \\ -0.3089 & -0.0059 & 0.0686 \end{bmatrix}}_{(X'X)^{-1}}$$` -- Example: `\(var(\hat{\beta}_1)=0.02389 \times 0.7325=0.0175\)` `\(cov(\hat{\beta}_1,\hat{\beta}_0)=0.02389 \times -0.3382=-0.00808\)` --- ### Variance - Where the hell do we get the `\(\sigma^2\)` from?! -- - Same as before: `$$\hat{\sigma}^2=\frac{\sum_i e_i^2}{n-p}$$` - Where `\(e_i\)` is fitted residual and `\(p\)` is number of parameters `\(p=k+1\)` - k is number of variables - +1 because of intercept - This is called mean squared error as well -- The easiest way to compute this sum is: `$$\sum_i e_i^2=\mathbf{e'}\mathbf{e}=(\mathbf{y-X\hat{\beta}})'(\mathbf{y-X\hat{\beta}})=\mathbf{y'y-\hat{\beta}'X'y}$$` --- #### Gauss Markov Theorem (Again) Assumptions - `\(E(u_i)=0\)` - `\(var(u_i)=\sigma^2\)` - `\(cov(u_i,u_j)=0\)` - `\(X\)` is full rank NO NEED FOR NORMALITY **Theorem:** OLS is .blue[BLUE:] Best, Linear, Unbiased Estimator - It has the lowest variance among linear and unbiased estimators -- - What's a linear estimator? - It's an estimator where `\(\beta\)` coefficients are linear functions of outcomes - Anything of the form `\(b=Cy\)` where C is p x n matrix. - So `\(b_1=c_{11}y_1+c_{12}y_2+...+c_{13}y_3\)` - Example `\(b_1=\frac{1}{n}y_1+...+\frac{1}{n}y_n\)` -- - How is OLS linear? `\(\hat{\beta}=Cy=\underbrace{(X'X)^{-1}X'}_{C}y\)` --- ### Categorical Variables in a Regression - Suppose we want to learn whether mode of work affects workers productivity. - Each worker can be in one of these 3 categories: - Fully at the office - Fully remote - Hybrid <div class="datatables html-widget html-fill-item" id="htmlwidget-33f20ff1ba179663bd3f" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-33f20ff1ba179663bd3f">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100],[93,75,108,115,75,88,107,115,125,99,104,114,121,125,73,100,91,121,118,77,113,84,74,126,79,109,80,71,101,75,87,95,105,118,82,98,97,114,113,120,123,127,103,101,80,99,85,72,108,102,78,87,105,80,76,96,91,121,86,71,122,90,108,103,93,81,96,108,101,110,114,99,93,70,70,130,76,95,113,115,122,106,115,121,107,118,71,95,122,109,125,80,89,118,129,89,70,130,79,73],["Fully remote","Fully remote","Hybrid","Hybrid","Hybrid","Hybrid","Fully remote","Fully at the office","Fully at the office","Hybrid","Fully at the office","Fully remote","Fully remote","Hybrid","Fully remote","Fully at the office","Hybrid","Fully at the office","Fully at the office","Hybrid","Fully remote","Fully at the office","Fully at the office","Fully remote","Fully remote","Fully remote","Fully remote","Fully at the office","Hybrid","Hybrid","Fully remote","Hybrid","Hybrid","Hybrid","Fully remote","Hybrid","Hybrid","Fully remote","Fully remote","Hybrid","Fully remote","Hybrid","Fully remote","Fully remote","Fully at the office","Fully remote","Hybrid","Fully at the office","Fully at the office","Hybrid","Fully remote","Fully at the office","Fully at the office","Hybrid","Hybrid","Hybrid","Fully at the office","Fully remote","Fully remote","Hybrid","Hybrid","Hybrid","Hybrid","Fully remote","Fully at the office","Hybrid","Fully remote","Fully at the office","Hybrid","Hybrid","Hybrid","Fully remote","Fully at the office","Fully remote","Hybrid","Fully at the office","Fully remote","Fully at the office","Fully remote","Fully remote","Hybrid","Fully remote","Fully remote","Fully remote","Fully remote","Fully remote","Hybrid","Fully at the office","Fully at the office","Hybrid","Hybrid","Hybrid","Hybrid","Fully remote","Fully at the office","Fully at the office","Hybrid","Fully remote","Fully remote","Hybrid"]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>WorkerID<\/th>\n <th>Productivity<\/th>\n <th>WorkMode<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":8,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[0,1]},{"name":"WorkerID","targets":0},{"name":"Productivity","targets":1},{"name":"WorkMode","targets":2}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[8,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> --- - How do we estimate the impact of categorical variable? - We turn it into a series of binary variables (or indicator variables)! `$$D_{i, Remote}=\begin{cases} 1 & WorkMode_i=Fully Remote \\ 0 & otherwise \end{cases}$$` `$$D_{i,Hybrid}=\begin{cases} 1 & WorkMode_i=Hybrid\\ 0 & otherwise \end{cases}$$` <div class="datatables html-widget html-fill-item" id="htmlwidget-1cc7f5a9fe246f7424ff" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-1cc7f5a9fe246f7424ff">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100],[112,124,108,76,125,111,79,115,127,95,88,86,83,104,115,110,103,119,116,71,93,73,118,125,103,121,105,110,101,116,124,119,74,70,73,122,105,89,128,105,102,93,89,119,73,92,121,112,111,91,103,78,117,123,82,116,106,127,80,102,122,122,71,129,99,93,95,76,80,94,91,95,88,104,79,127,73,92,128,109,74,95,96,100,97,107,78,124,104,103,77,116,99,117,73,119,86,87,109,127],["Fully at the office","Hybrid","Hybrid","Fully at the office","Fully remote","Fully at the office","Fully remote","Fully remote","Fully at the office","Hybrid","Fully at the office","Fully remote","Fully remote","Fully at the office","Fully remote","Fully at the office","Fully at the office","Fully remote","Fully remote","Fully remote","Fully remote","Hybrid","Fully at the office","Fully at the office","Fully at the office","Fully at the office","Hybrid","Fully remote","Fully at the office","Fully at the office","Hybrid","Hybrid","Fully at the office","Fully at the office","Fully remote","Hybrid","Fully at the office","Fully at the office","Hybrid","Fully at the office","Fully remote","Fully remote","Hybrid","Fully at the office","Hybrid","Fully at the office","Hybrid","Hybrid","Fully at the office","Hybrid","Hybrid","Hybrid","Hybrid","Hybrid","Fully remote","Hybrid","Fully remote","Hybrid","Fully remote","Fully at the office","Fully at the office","Hybrid","Fully at the office","Fully remote","Fully at the office","Fully remote","Fully remote","Fully at the office","Fully at the office","Fully remote","Hybrid","Fully at the office","Fully remote","Fully at the office","Fully remote","Hybrid","Fully remote","Fully remote","Hybrid","Fully at the office","Fully remote","Hybrid","Hybrid","Fully at the office","Hybrid","Fully at the office","Fully at the office","Hybrid","Fully at the office","Fully at the office","Hybrid","Fully at the office","Fully at the office","Fully at the office","Fully remote","Fully remote","Fully remote","Fully remote","Hybrid","Hybrid"],[1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,1,0,0,0,0,0,1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0,0,0,0,0,1,0,0,0,1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0],[0,0,0,0,1,0,1,1,0,0,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,0,0,0,1,0,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0],[0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,1,0,0,0,1,0,0,1,0,0,0,1,0,1,0,1,1,0,1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,1,0,0,1,1,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,1,1]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>WorkerID<\/th>\n <th>Productivity<\/th>\n <th>WorkMode<\/th>\n <th>WorkModeFully.at.the.office<\/th>\n <th>WorkModeFully.remote<\/th>\n <th>WorkModeHybrid<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":6,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[0,1,3,4,5]},{"name":"WorkerID","targets":0},{"name":"Productivity","targets":1},{"name":"WorkMode","targets":2},{"name":"WorkModeFully.at.the.office","targets":3},{"name":"WorkModeFully.remote","targets":4},{"name":"WorkModeHybrid","targets":5}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[6,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> -- - For each person, only one of these dummies is equal to 1! --- - We will add these dummies into a regression, but not all of them! - If we have m categories, we will add m-1 dummies. Why? `$$y_i=\beta_0+\beta_1D_{i1}+\beta_2D_{i2}+...+\beta_{m-1}D_{im-1}+u_i$$` - In our Example: `$$y_i=\beta_0+\beta_1D_{i,Hybrid}+\beta_2D_{i,Remote}+u_i$$` -- - Because otherwise X would not be full rank! <div class="math"> \[ \begin{array}{cc} \text{Full Rank Matrix:} & \text{Matrix Not of Full Rank:} \\ \left[\begin{array}{ccc} 1 & 1 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 1 \end{array}\right] & \left[\begin{array}{cccc} 1 & 1 & 0 & 0\\ 1 & 0 & 0 & 1\\ 1 & 0 & 1 & 0 \end{array}\right] \end{array} \] </div> -- - Intuitively, if I know that the values of `\(D_{i,Hybrid}\)` and `\(D_{i,Remote}\)`, I know the value of `\(D_{i,Office}\)` - Ex: if they don't work hybrid and don't work remote, I know they work at the office - So including it does not bring any new information --- - R automatically transform categorical variable to dummies and excludes one of them ``` r # Fit a linear regression model lm_model <- lm(Productivity ~ WorkMode, data = d) # Display the summary of the linear regression model summary(lm_model) ``` ``` ## ## Call: ## lm(formula = Productivity ~ WorkMode, data = d) ## ## Residuals: ## Min 1Q Median 3Q Max ## -34.774 -12.636 0.946 14.410 34.667 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 101.590 2.695 37.697 <2e-16 *** ## WorkModeFully remote -7.256 4.087 -1.775 0.079 . ## WorkModeHybrid 6.184 4.050 1.527 0.130 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 16.83 on 97 degrees of freedom ## Multiple R-squared: 0.09125, Adjusted R-squared: 0.07251 ## F-statistic: 4.87 on 2 and 97 DF, p-value: 0.009652 ``` --- ### Interpretation of Coefficients - Coefficient on a dummy `\(D_1\)` tells us by how much `\(y\)` changes when we change category from the excluded one to the category 1. - In our example - Excluded category is: work fully at the office - this is our comparision group - `\(\beta_{hybrid}=6.184\)`: employees working in hybrid mode have on average 6.184 higher productivity score compared to the ones working at the office - `\(\beta_{remote}=-7.256\)`: employees working in fully remotely have on average 7.256 lower productivity score compared to the ones working at the office - The t-test on these coefficients tells us whether these differences in means across categories are significant! - Bottom line: the coefficients on the dummies show the average difference between `\(y\)` in that category compared to the excluded category (holding everything else unchanged) --- ### Example Suppose we have a categorical variable representing education level. 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Error t value Pr(>|t|) ## (Intercept) 70342 3125 22.509 < 2e-16 *** ## EducationPhD 14639 4008 3.652 0.000424 *** ## EducationMaster 22303 4157 5.365 5.59e-07 *** ## EducationBachelor 16993 4273 3.977 0.000135 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 13980 on 96 degrees of freedom ## Multiple R-squared: 0.2401, Adjusted R-squared: 0.2164 ## F-statistic: 10.11 on 3 and 96 DF, p-value: 7.517e-06 ``` --- ### Interactions Consider a regression: $$\text{Purchases}_i=\beta_0+\beta_1\text{TikTok ads}_i+\beta_2\text{Age<25}_i+u_i $$ - Where tiktok ads is how much the company spends on tiktok ads - Age<25 is a dummy variable that is 1 if the person is younger than 25 - Purchases is how much person i spent on clothes from a company. <img src="data:image/png;base64,#C_5_slides_a_files/figure-html/unnamed-chunk-19-1.png" width="100%" /> --- ### Interactions ``` ## ## Call: ## lm(formula = sales ~ tiktok_ad_spend + young, data = data) ## ## Residuals: ## Min 1Q Median 3Q Max ## -141.883 -50.858 0.821 47.398 145.724 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 64.73544 10.12217 6.395 1.14e-09 *** ## tiktok_ad_spend 0.56359 0.03244 17.373 < 2e-16 *** ## young1 208.29233 8.89550 23.415 < 2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 62.69 on 197 degrees of freedom ## Multiple R-squared: 0.8149, Adjusted R-squared: 0.813 ## F-statistic: 433.7 on 2 and 197 DF, p-value: < 2.2e-16 ``` --- ### Interactions - Do ads on tik tok affect in the same way older and younger people? - In other words: one additional dollar on tik-tok ads increases purchases more if you are young? -- - We want allow the coefficient on ads to differ by age group. <img src="data:image/png;base64,#C_5_slides_a_files/figure-html/unnamed-chunk-21-1.png" width="100%" /> --- ### Interactions - Run the regression: $$\text{Purchases}_i=\beta_0+\beta_1\text{TikTok ads}_i+\beta_2\text{Age<25}_i+\beta_3\text{TikTok ads}_i*\text{Age<25}_i+u_i $$ -- - What's the coefficient on ads when you are older `\(Age<25_i=0\)`? `$$\begin{align*} & \text{Purchases}_i=\beta_0+\beta_1\text{TikTok ads}_i+\beta_2*0+\beta_3\text{TikTok ads}_i*0 +u_i \\ & \text{Purchases}_i=\beta_0+\beta_1\text{TikTok ads}_i+u_i \end{align*}$$` -- - What's the coefficient on ads when you are younger `\(Age<25_i=1\)`? `$$\begin{align*} & \text{Purchases}_i=\beta_0+\beta_1\text{TikTok ads}_i+\beta_2*1+\beta_3\text{TikTok ads}_i*1 +u_i \\ & \text{Purchases}_i=\beta_0+\beta_2+(\beta_1+\beta_3)\text{TikTok ads}_i+u_i \end{align*}$$` -- We can estimate `\(\beta_3\)` and it will tell us by how much bigger is the coefficient on ads for young compared to the coefficient on ads for old. - `\(\beta_1\)` is the slope for old - `\(\beta_1+\beta_3\)` is the slope for young - `\(\beta_3\)` is the difference in slopes, which we can test like other coefficients --- ``` ## ## Call: ## lm(formula = sales ~ tiktok_ad_spend * young, data = data) ## ## Residuals: ## Min 1Q Median 3Q Max ## -103.867 -25.348 -1.574 24.928 110.078 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 150.44290 8.13287 18.498 < 2e-16 *** ## tiktok_ad_spend 0.22153 0.02865 7.733 5.38e-13 *** ## young1 29.35544 11.85879 2.475 0.0142 * ## tiktok_ad_spend:young1 0.70583 0.04115 17.152 < 2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 39.74 on 196 degrees of freedom ## Multiple R-squared: 0.926, Adjusted R-squared: 0.9249 ## F-statistic: 817.6 on 3 and 196 DF, p-value: < 2.2e-16 ``` - One additional dollar on ads increases purchases among old by 0.222 dollars - One additional dollar on ads increases purchases among young by 0.222+0.706=0.928 dollars --- ### Interactions - More generally, we can rewrite a regression: `$$y_i=\beta_0+\beta_1x_{i1}+\beta_2x_{i2}+\beta_3x_{i1}*x_{i2}+u_i$$` - As `$$y_i=\beta_0+(\beta_1+\beta_3x_{i2})x_{i1}+\beta_2x_{i2}+u_i$$` -- - `\(\beta_3\)` answers the following question: - If I increase `\(x_{i2}\)` by one, by how much the coefficient on `\(x_{i1}\)` changes? - If `\(x_{i2}\)` is categorical: by how much the effect of `\(x_{i1}\)` changes when `\(x_{i2}=1\)` compared to when `\(x_{i2}=0\)`? -- - Suppose `\(y\)` is mortality, `\(x_1\)` is temperature, `\(x_2\)` is age - What would you expect `\(\beta_3\)` to be? --- ### Interactions Exam example  --- ### Interactions - Suppose you want to know who benefits the most from working from home. You collect survey data for each employee on the job satisfaction, whether they work in the office or from home, and the distance between the office and home -- - Who do you think benefits most from working from home? -- - How would you test this? -- `$$\text{Satisfaction}_i=\beta_0+\beta_1\text{WFH}_i+\beta_2\text{Distance}_i+\beta_3\text{WFH}_i*\text{Distance}_i +u_i$$` -- - What's the interpretation of `\(\beta_3\)`? -- - By how much the effect of working from home on satisfaction changes when we increase distance by one unit (km) -- - Which sign do you expect `\(\beta_3\)` to have? --- --- ### Example When choosing a partner, do women or men care more about physical attractivness? -- ``` ## ## Call: ## lm(formula = Like ~ Attractive * gender, data = SpeedDating2) ## ## Residuals: ## Min 1Q Median 3Q Max ## -4.7963 -0.6257 0.0914 0.7890 4.7155 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 2.69922 0.26783 10.078 <2e-16 *** ## Attractive 0.58530 0.04082 14.338 <2e-16 *** ## genderM -0.78822 0.40226 -1.959 0.0506 . ## Attractive:genderM 0.12864 0.05953 2.161 0.0312 * ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 1.292 on 540 degrees of freedom ## (8 observations deleted due to missingness) ## Multiple R-squared: 0.4732, Adjusted R-squared: 0.4703 ## F-statistic: 161.7 on 3 and 540 DF, p-value: < 2.2e-16 ``` --- ### Goodness of fit - We can use again the R square to measure the goodness of fit. `$$\small R^2=1-\frac{\sum(y_i-\hat{y}_i)^2}{\sum(y_i-\bar{y}_i)^2}$$` -- - However, there is one problem with it. - Even if we add variables unrelated to `\(y\)`, the `\(R^2\)` would typically still increase by a bit -- - Even if in population there is 0 relationship with this variable, our sample is small so we will never get exactly 0 relationship -- - Sampling noise will make coefficient slightly positive or negative -- - So the increase in `\(R^2\)` will reflect that noise in our sample -- - The more coefficients we include, the higher `\(R^2\)` - We can adjust it, by accounting for the number of parameters used -- `$$\small R_{Adj}^2=1-\frac{\sum(y_i-\hat{y}_i)^2/(n-p)}{\sum(y_i-\bar{y}_i)^2/(n-1)}$$` -- - More parameters -> `\(\downarrow(n-p)\rightarrow\uparrow\sum(y_i-\hat{y}_i)^2/(n-p)\rightarrow\downarrow R_{Adj}^2\)` - So it balances off the mechanical effect of higher `\(R^2\)` due to more regressors --- ``` ## ## Call: ## lm(formula = Duration ~ Occupancy + EDAD, data = Sample_urg[Sample_urg$SEXO != ## "NO ESPECIFICADO", ]) ## ## Residuals: ## Min 1Q Median 3Q Max ## -773.65 -26.61 -17.27 -0.57 1252.75 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 23.23422 2.48416 9.353 < 2e-16 *** ## Occupancy 3.70354 0.10090 36.705 < 2e-16 *** ## EDAD 0.20626 0.06747 3.057 0.00225 ** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 98.99 on 4995 degrees of freedom ## Multiple R-squared: 0.2169, Adjusted R-squared: 0.2166 ## F-statistic: 691.8 on 2 and 4995 DF, p-value: < 2.2e-16 ``` --- ``` ## ## Call: ## lm(formula = Duration ~ Occupancy + EDAD + Random_var, data = Sample_urg[Sample_urg$SEXO != ## "NO ESPECIFICADO", ]) ## ## Residuals: ## Min 1Q Median 3Q Max ## -773.18 -26.60 -17.26 -0.47 1253.34 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 23.09257 2.49595 9.252 < 2e-16 *** ## Occupancy 3.70288 0.10091 36.693 < 2e-16 *** ## EDAD 0.20588 0.06747 3.051 0.00229 ** ## Random_var 0.02755 0.04680 0.589 0.55616 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 99 on 4994 degrees of freedom ## Multiple R-squared: 0.217, Adjusted R-squared: 0.2165 ## F-statistic: 461.2 on 3 and 4994 DF, p-value: < 2.2e-16 ``` - Adding random variable increased `\(R^2\)` but decreased `\(R^2_{Adj}\)` --- layout: false class: inverse, middle # Statistical Properties of OLS --- ### Inference - Let's add the assumption that errors are normally distributed: $$ \mathbf{u} \sim N(0,\sigma I) $$ Which means that: $$y \sim N(X\beta,\sigma I) $$ - With inference we can: - Do hypothesis testing on single coefficients, ex: `\(H_0: \beta_2=0\)` - Find confidence intervals for a single coefficients - Do hypothesis testing on multiple coefficients: ex: `\(H_0: \beta_1=\beta_2\)` --- ### Test for a Single Coefficient Under the above assumptions: `$$\hat{\beta}\sim N(\beta, \sigma\sqrt{(X'X)^{-1}})$$` And `$$\hat{\beta_j}\sim N(\beta_j, \sigma\sqrt{(X'X)^{-1}_{j+1,j+1}})$$` -- Normalizing we get that: `$$\frac{\hat{\beta}_j-\beta_j}{s\sqrt{(X'X)^{-1}_{j+1,j+1}}} \sim t_{n-p}$$` - This test statistic has student t distribution with n-p degrees of freedom - Because the `\(\frac{s^2(n-p)}{\sigma^2} \sim \chi_{n-p}\)` - Where p is the number of parameters (coefficients) - `\(p=k+1\)`: k regressors and 1 intercept --- ### Test for a single coefficient Suppose: - `\(H_0: \beta_j=\beta_{j0}\)` - `\(H_A: \beta_j \neq \beta_{j0}\)` Then, we use test statistic: `$$t_{test}=\frac{\hat{\beta}_j-\beta_{j0}}{s\sqrt{(X'X)^{-1}_{j+1,j+1}}}$$` And we reject if `\(t_{test}>t_{\alpha/2,n-p}\)` or `\(t_{test}<-t_{\alpha/2,n-p}\)` Where `\(t_{\alpha/2,n-p}\)` is `\(1-\alpha/2\)` quantile of student t with n-p degrees of freedom -- **NOTE:** This is a test for `\(\beta_j\)` given all other regressors. It's not the same as the test statistic with only one regressor! --- ### Example Suppose: - `\(H_0: \beta_{Age}=0\)` - `\(H_A: \beta_{Age} \neq 0\)` ``` ## ## Call: ## lm(formula = Duration ~ Occupancy + EDAD, data = Sample_urg) ## ## Residuals: ## Min 1Q Median 3Q Max ## -773.65 -26.61 -17.27 -0.57 1252.75 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 23.23422 2.48416 9.353 < 2e-16 *** ## Occupancy 3.70354 0.10090 36.705 < 2e-16 *** ## EDAD 0.20626 0.06747 3.057 0.00225 ** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 98.99 on 4995 degrees of freedom ## Multiple R-squared: 0.2169, Adjusted R-squared: 0.2166 ## F-statistic: 691.8 on 2 and 4995 DF, p-value: < 2.2e-16 ``` --- ## Example  --- ### Confidence Interval for a Single Coefficient We can also use this distribution to construct confidence intervals: An interval for `\(\beta_j\)` with confidence level `\(1-\alpha\)` is: `$$\begin{align*} CI_{1-\alpha} & =\{\hat{\beta_j}-t_{\alpha/2,n-p}SE(\hat{\beta}_j),\hat{\beta_j}+t_{\alpha/2,n-p}SE(\hat{\beta}_j)\} \\ & =\{\hat{\beta_j}-t_{\alpha/2,n-p}s\sqrt{(X'X)^{-1}_{j+1,j+1}},\hat{\beta_j}+t_{\alpha/2,n-p}s\sqrt{(X'X)^{-1}_{j+1,j+1}}\} \end{align*}$$` -- **Intepretation:** - We are `\(1-\alpha\)` % confident that the true parameter is within this CI - If we take repeated samples, `\(1-\alpha\)` % of such constructed confidence intervals would contain true `\(\beta\)` --- ### Example: For our age coefficient we had: - `\(\hat{\beta}_{Age}=0.206\)` - `\(SE(\hat{\beta})=0.067\)` - Our `\(n=5000\)` so we can use normal approximation -- So 95% CI for `\(\beta_{Age}\)` is: `$$\begin{align*} CI_{1-\alpha} & =\{\hat{\beta_j}-t_{\alpha/2,n-p}SE(\hat{\beta}_j),\hat{\beta_j}+t_{\alpha/2,n-p}SE(\hat{\beta}_j)\} \\ & =\{0.206-1.96*0.067,0.206+1.96*0.067\} \\ & =\{0.075,0.337\} \end{align*}$$` -- - Note that the CI does not contain 0 - What does it imply for hypothesis testing with `\(H_0: \beta_{age}=0\)`? --- ### CI for mean response .pull-left[ Suppose that we want an average prediction for individuals with these characteristics: `$$\mathbf{x_0}=\begin{bmatrix} 1 \\ x_{01} \\ x_{02} \\ \vdots \\ x_{0k} \\ \end{bmatrix}$$` Ex: What's average wait time if $x_{01}=10 (10 other people are there) and are age 52 `\(x_{02}=52\)` How accurate is our prediction? `$$\hat{y}_0=\mathbf{x_0}'\hat{\beta}$$` ] .pull-right[ The prediction is unbiased: `\(E(\hat{y}_0)=\mathbf{x_0}'\beta\)` and it's variance is: `$$\begin{align*} var(\hat{y}_0)& =var(\mathbf{x_0}'\hat{\beta}) \\ & =\mathbf{x_0}'var(\hat{\beta})\mathbf{x_0} \\ &=\sigma^2\mathbf{x_0}'(\mathbf{X}'\mathbf{X})^{-1}\mathbf{x_0} \end{align*}$$` So it's distribution is: `\(\hat{y}_0 \sim N(\mathbf{x_0}'\beta, \sqrt{\sigma^2\mathbf{x_0}'(\mathbf{X}'\mathbf{X})^{-1}\mathbf{x_0}})\)` Hence: `\(CI_{1-\alpha}=\{\hat{y}_0\pm t_{n-p,\frac{\alpha}{2}}\sqrt{\sigma^2\mathbf{x_0}'(\mathbf{X}'\mathbf{X})^{-1}\mathbf{x_0}}\}\)` ] --- ### Example What's the 95% CI for average wait time when there is 10 people at the Urgent Care `\(x_{Occupancy}=10\)` for a person who is of age 52 `\(x_{age}=52\)` ? - What do we need to answer this question? - `\(\hat{\beta}=\{\hat{\beta_0}, \hat{\beta}_{Occupancy}, \hat{\beta}_{age}\}=\{23.236, 3.7, 0.2\}\)` - `\(\hat{\sigma}=98.97\)` - `\((X'X)^{-1}=\)` ``` ## (Intercept) Occupancy EDAD ## (Intercept) 6.297395e-04 -5.434372e-06 -1.331625e-05 ## Occupancy -5.434372e-06 1.038940e-06 -5.573690e-08 ## EDAD -1.331625e-05 -5.573690e-08 4.644967e-07 ``` --- ### Example - Prediction: `\(\hat{y_0}=23.236*1+3.7*10+0.2*52=70.636\)` -- - `\(\sqrt{\mathbf{x_0}'(\mathbf{X}'\mathbf{X})^{-1}\mathbf{x_0}}=\sqrt{[1, 10, 52](\mathbf{X}'\mathbf{X})^{-1}[1, 10, 52]'}=0.021\)` - Standard Deviation: `\(SE(\hat{y_0})=\sqrt{\sigma^2\mathbf{x_0}'(\mathbf{X}'\mathbf{X})^{-1}\mathbf{x_0}}=2.07837\)` -- `$$CI_{95}=\{70.636 \pm 1.96*2.07837\} \approx \{67, 75\}$$` --- ### Exanmple Or simply in R: ``` r lm_model <- lm(Duration ~ Occupancy+EDAD, data = Sample_urg) new_data<- data.frame(Occupancy= c(10), EDAD=52) predict(lm_model, newdata = new_data, interval = "confidence", level = 0.95, se.fit=TRUE) ``` ``` ## $fit ## fit lwr upr ## 1 70.9952 66.93326 75.05714 ## ## $se.fit ## [1] 2.071955 ## ## $df ## [1] 4995 ## ## $residual.scale ## [1] 98.99182 ``` --- ### CI for new observation ** Reminder **: - When we look at average response, `\(u_i\)` doesn't play a role (because on average errors are 0) - When we look at a single observation, `\(u_i\)` matters, so it increases the variance of prediction error So variance is now the previous variance plus the variance of `\(u_i\)` `$$\begin{align*}var(y_0-\hat{y}_0)& =var(x_0\beta+u_i-x_0\hat{\beta}) \\ & =var(u_0)+var(x_0\hat{\beta}) \\ & =\sigma^2+\sigma^2\mathbf{x_0}'(\mathbf{X}'\mathbf{X})^{-1}\mathbf{x_0}\end{align*}$$` So the confidence interval for a single observation is slightly wider: `\(CI_{1-\alpha}=\{\hat{y}_0\pm t_{n-p,\frac{\alpha}{2}}\sqrt{\sigma^2(1+\mathbf{x_0}'(\mathbf{X}'\mathbf{X})^{-1}\mathbf{x_0})}\}\)` We are less certain about predicting outcome for a single person, compared to average outcome among namy people. --- ### Testing for the significance of the regression - Does our model helps to explain any variation in `\(y_i\)`? - `\(\small H_0: \beta_1=\beta_2=...\beta_k=0\)` - `\(\small H_A: \beta_j \neq 0\)` for at least one `\(j\)` -- - It's the same procedure as before! - **Explained variation** should be large compared to **unexplained variation** if the model works -- - We can again do the decomposition in SST, SSR, and SSE: - `\(SS_T\)` is total sum of squares `\(\sum_i(y_i-\bar{y})^2\)`, .blue[n-1 DoF] - `\(SS_R\)` is regression sum of squares `\(\sum_i(\hat{y_i}-\bar{y})^2\)`, .blue[k DoF] - `\(SS_E\)` is residual error sum of squares `\(\sum_i(y_i-\hat{y_i})^2\)`, .blue[n-k-1 DoF] <table> <thead> <tr> <th style="text-align: left;">Source</th> <th style="text-align: center;">Sum of Squares</th> <th style="text-align: center;">Degrees of Freedom</th> <th style="text-align: center;">DoF</th> </tr> </thead> <tbody> <tr> <td style="text-align: left;">Regression</td> <td style="text-align: center;">13557462</td> <td style="text-align: center;">2</td> <td style="text-align: center;">k</td> </tr> <tr> <td style="text-align: left;">Residual Error</td> <td style="text-align: center;">48947909</td> <td style="text-align: center;">4995</td> <td style="text-align: center;">n-k-1</td> </tr> <tr> <td style="text-align: left;">Total</td> <td style="text-align: center;">62505371</td> <td style="text-align: center;">4997</td> <td style="text-align: center;">n-1</td> </tr> </tbody> </table> --- ### Testing for the significance of the regression F-stat and its distribution under the null `$$F_{stat}=\frac{SSR/(k)}{SSE/(n-k-1)} \sim \underbrace{F_{k,n-k-1}}_{\text{Dist under } H_0}$$` --- ### Testing for the significance of the regression Alternative way to think about it: - `\(\small H_0: y=\beta_0+u\)` restricted model ( `\(x_1\)` and `\(x_2\)` don't matter) - `\(\small H_A: y=\beta_0+\beta_1x_{1}+\beta_2x_{2}+u\)` If `\(H_A\)` is true, unrestricted model should explain more of `\(y\)` `$$\small F_{stat}=\frac{SSR/(k)}{SSE/(n-k-1)}=\frac{\frac{\overbrace{SSR_{H_A}-SSR_{H_0}}^{\text{Extra Sum of Squares}}}{k-(k_0)}}{\frac{SSE_{H_A}}{n-k-1}}=\frac{\frac{\overbrace{SSE_{H_0}-SSE_{H_A}}^{\text{Extra Sum of Squares}}}{k-(k_0)}}{\frac{SSE_{H_A}}{n-k-1}}$$` -- - `\(\small SSR_{H_A}\)` is the regression sum of square from unrestricted model with `\(\small k\)` degrees of freedom (2) - `\(\small SSR_{H_0}\)` is the regression sum of squres from the restricted model wiht `\(\small k_0\)` degrees of freedom (0) - it's the number of regressors in restricted model - `\(\small SSE_{H_A}\)` is the residual sum of squres in the unrestricted model --- ### Testing for the significance of the regression ``` r linearHypothesis(lm_model, c("Occupancy=0", "EDAD=0")) ``` ``` ## ## Linear hypothesis test: ## Occupancy = 0 ## EDAD = 0 ## ## Model 1: restricted model ## Model 2: Duration ~ Occupancy + EDAD ## ## Res.Df RSS Df Sum of Sq F Pr(>F) ## 1 4997 62505371 ## 2 4995 48947909 2 13557462 691.75 < 2.2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ``` --- ## Example  --- ### Adding a coefficient (skip) - We can use the above logic to test how much more we can explain by including one more coeffcient -- - Suppose we want to compare a regression model with only Occupancy vs both Occupancy and age - `\(\small H_0: y=\beta_0+\beta_1Occupancy+u\)` restricted model - `\(\small H_A: y=\beta_0+\beta_1Occupancy+\beta_2Age+u\)` unrestricted model `$$\small F_{2}=\frac{\frac{\overbrace{SSR_{H_A}-SSR_{H_0}}^{\text{Extra Sum of Squares}}}{k-(k_0)}}{\frac{SSE_{H_A}}{n-k-1}}=\frac{\frac{\overbrace{SSE_{H_0}-SSE_{H_A}}^{\text{Extra Sum of Squares}}}{k-(k_0)}}{\frac{SSE_{H_A}}{n-k-1}} \sim \underbrace{F_{k-k_0, n-k-1}}_{\text{Dist under } H_0}$$` - In our case `\(k=2\)` and `\(k_0=1\)`, so the null distribution is `\(F_{1, n-3}\)` --- ### Adding a coefficient (skip) ``` ## Analysis of Variance Table ## ## Response: Duration ## Df Sum Sq Mean Sq F value Pr(>F) ## Occupancy 1 13465872 13465872 1374.1553 < 2.2e-16 *** ## EDAD 1 91590 91590 9.3465 0.002246 ** ## Residuals 4995 48947909 9799 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ``` - **Sequential testing:** - Occupancy - `\(F_1\)` is the additional effect of including Occupancy to a model without any regressors - `\(\small H_0: y=\beta_0+u\)` restricted model - `\(\small H_A: y=\beta_0+\beta_1Occupancy+u\)` unrestricted model - EDAD - `\(F_2\)` is the additional effect of including Age once we already have Occupancy in the model - `\(\small H_0: y=\beta_0+\beta_1Occupancy+u\)` restricted model - `\(\small H_A: y=\beta_0+\beta_1Occupancy+\beta_2Age+u\)` unrestricted model --- ### Adding a coefficient (skip) - `\(F_k\)` (last coef) is equivalent to `\(t_k^2\)` in our full model - But `\(F_1\)` is not equivalent to `\(t_1^2\)` in our full model ``` ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 23.2342216 2.48416121 9.352944 1.255500e-20 ## Occupancy 3.7035443 0.10090081 36.704802 2.396768e-261 ## EDAD 0.2062604 0.06746688 3.057209 2.245903e-03 ``` ``` ## Analysis of Variance Table ## ## Response: Duration ## Df Sum Sq Mean Sq F value Pr(>F) ## Occupancy 1 13465872 13465872 1374.1553 < 2.2e-16 *** ## EDAD 1 91590 91590 9.3465 0.002246 ** ## Residuals 4995 48947909 9799 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ``` --- ### Adding a coefficient (skip) - Why reordering variables changes `\(F_{stats}\)`? ``` ## Analysis of Variance Table ## ## Response: Duration ## Df Sum Sq Mean Sq F value Pr(>F) ## Occupancy 1 13465872 13465872 1374.1553 < 2.2e-16 *** ## EDAD 1 91590 91590 9.3465 0.002246 ** ## Residuals 4995 48947909 9799 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ``` ``` ## Analysis of Variance Table ## ## Response: Duration ## Df Sum Sq Mean Sq F value Pr(>F) ## EDAD 1 355320 355320 36.259 1.851e-09 *** ## Occupancy 1 13202142 13202142 1347.243 < 2.2e-16 *** ## Residuals 4995 48947909 9799 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ``` -- - Because it changes which regressors we already have in the model -- - Do squares always add up to the same thing? --- ### Testing multiple coefficients (skip) Suppose we have a model with three predictors `$$\small y=\beta_0+\beta_1Occupancy+\beta_2Age+\beta_3Male+u$$` We can test for a subset of predictors, for example if Age and Sex matter - `\(H_0:\beta_2=\beta_3=0 \rightarrow y=\beta_0+\beta_1Occupancy+u\)` - `\(H_A:\beta_2\neq 0 \text{ or } \beta_3\neq0 \\ \rightarrow y=\beta_0+\beta_1Occupancy+\beta_2Age+\beta_3Male+u\)` `$$\small F_{test}=\frac{\frac{\overbrace{SSR_{H_A}-SSR_{H_0}}^{\text{Extra Sum of Squares}}}{3-1}}{\frac{SSE_{H_A}}{n-3-1}}=\frac{\frac{\overbrace{SSE_{H_0}-SSE_{H_A}}^{\text{Extra Sum of Squares}}}{3-1}}{\frac{SSE_{H_A}}{n-3-1}} \sim F_{2, n-4}$$` --- ``` ## ## Linear hypothesis test: ## EDAD = 0 ## SEXOMASCULINO = 0 ## ## Model 1: restricted model ## Model 2: Duration ~ Occupancy + EDAD + SEXO ## ## Res.Df RSS Df Sum of Sq F Pr(>F) ## 1 4996 49039499 ## 2 4994 48728235 2 311264 15.95 1.244e-07 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ``` --- ### Testing for multiple coefficients A cool thing about the regression is that we can test relationships between the coefficients: **For example**: - Is the impact of additional year of education the same as impact of additional year of work experience in a regression: `$$income_i=\beta_0+\beta_1\text{education}_i+\beta_2\text{experience}_i+u_i$$` - That corresponds to null hypothesis `\(H_0: \beta_1=\beta_2\)` or `\(H_0: \beta_1-\beta_2=0\)` **Another Example**: - Suppose that employees can go through a sales training, and/or get a better office (these are binary variables). We want to evaluate impact of these measures on their sales: `$$Sales_i=\beta_0+\beta_1\text{training}_i+\beta_2\text{office}_i+u_i$$` - We wonder if giving an employee both of them would increase sales by more than 100: `\(H_A: \beta_1+\beta_2>100\)` --- ### Relationships between coefficients Suppose we have a model: `$$y=\beta_0+\beta_1x_1+\beta_2x_2+...\beta_kx_k+u$$` -- - We want to test if the difference between impact of `\(x_1\)` and `\(x_2\)` is equal to c **Hypothesis** - `\(H_0: \beta_1-\beta_2=c\)` - `\(H_A: \beta_1-\beta_2\neq c\)` -- - Special case: `\(c=0\)` => testing equality `\(\beta_1=\beta_2\)` -- **Test statistic and its distribution under the null** `$$T_{test}=\frac{\hat{\beta_1}-\hat{\beta_2}-c}{SE(\hat{\beta_1}-\hat{\beta_2})}=\frac{\hat{\beta_1}-\hat{\beta_2}-c}{\sqrt{var(\hat{\beta_1})+var(\hat{\beta_2})-2cov(\hat{\beta_1},\hat{\beta_2})}}\sim t_{n-k-1}$$` - Calculate p-value as `\(2P(t_{n-k-1}>|T_{test}|)\)` --- ### Relationships between coefficients - In the same way we can test whether one coefficient is larger than another by some amount **Hypothesis** - `\(H_0: \beta_1-\beta_2=c\)` - `\(H_A: \beta_1-\beta_2> c\)` -- - Special case: `\(c=0\)` => testing inequality `\(\beta_1>\beta_2\)` -- **Test statistic and its distribution under the null** `$$T_{test}=\frac{\hat{\beta_1}-\hat{\beta_2}-c}{SE(\hat{\beta_1}-\hat{\beta_2})}=\frac{\hat{\beta_1}-\hat{\beta_2}-c}{\sqrt{var(\hat{\beta_1})+var(\hat{\beta_2})-2cov(\hat{\beta_1},\hat{\beta_2})}}\sim t_{n-k-1}$$` -- - Calculate p-value as `\(P(t_{n-k-1}>T_{test})\)` -- - If alternative is `\(H_A: \beta_1-\beta_2 < c\)`, then `\(P(t_{n-k-1}<T_{test})\)` --- ### Example - Test if one more person at the hospital has larger effect than being one year older ``` ## ## Call: ## lm(formula = Duration ~ Occupancy + EDAD + SEXO, data = Sample_urg) ## ## Coefficients: ## (Intercept) Occupancy EDAD SEXOMASCULINO ## 18.5463 3.6803 0.2047 13.8988 ``` ``` ## (Intercept) Occupancy EDAD SEXOMASCULINO ## (Intercept) 7.12075807 -0.0481948400 -0.1296097547 -2.8941142167 ## Occupancy -0.04819484 0.0101612131 -0.0005422531 -0.0143206543 ## EDAD -0.12960975 -0.0005422531 0.0045323658 -0.0009536548 ## SEXOMASCULINO -2.89411422 -0.0143206543 -0.0009536548 8.5804013484 ``` --- ### Example - Hypotheses: - `\(H_0: \beta_{O}=\beta_{A}\)` - `\(H_A: \beta_{O}>\beta_{A}\)` -- - Calculate the test statistic `$$\small T_{test}=\frac{\beta_{O}-\beta_{A}}{\sqrt{var(\hat{\beta_O})+var(\hat{\beta_A})-2cov(\hat{\beta_O},\hat{\beta_A})}}=\frac{3.6803- 0.2047}{\sqrt{0.01+0.0045-2*(-0.00054)}}=27.84$$` -- - Calculate p-value `\(P-value=P(t_{n-k-1}>T_{test})=P(t_{4994}>27.84) \approx0\)` -- Conclusion - we reject that impact of one more year is smaller or equal to the impact of one more person --- ### Sum of coefficients Suppose we have a model: `$$y=\beta_0+\beta_1x_1+\beta_2x_2+...\beta_kx_k+u$$` -- - We want to test if the sum of impact of `\(x_1\)` and `\(x_2\)` is equal to c **Hypothesis** - `\(H_0: \beta_1+\beta_2=c\)` - `\(H_A: \beta_1+\beta_2\neq c\)` -- **Test statistic and its distribution under the null** `$$T_{test}=\frac{\hat{\beta_1}+\hat{\beta_2}-c}{SE(\hat{\beta_1}+\hat{\beta_2})}=\frac{\hat{\beta_1}+\hat{\beta_2}-c}{\sqrt{var(\hat{\beta_1})+var(\hat{\beta_2})+2cov(\hat{\beta_1},\hat{\beta_2})}}\sim t_{n-k-1}$$` -- - Calculate p-value as `\(2P(|t_{n-k-1}|>T_{test})\)` - If `\(H_A: \beta_1+\beta_2 < c\)`, then `\(P(t_{n-k-1}<T_{test})\)` - If `\(H_A: \beta_1+\beta_2 > c\)`, then `\(P(t_{n-k-1}>T_{test})\)` --- ### Example - Test if the total impact of increasing Occupancy by one person and being male is larger than 17 ``` ## ## Call: ## lm(formula = Duration ~ Occupancy + EDAD + SEXO, data = Sample_urg) ## ## Coefficients: ## (Intercept) Occupancy EDAD SEXOMASCULINO ## 18.5463 3.6803 0.2047 13.8988 ``` ``` ## (Intercept) Occupancy EDAD SEXOMASCULINO ## (Intercept) 7.12075807 -0.0481948400 -0.1296097547 -2.8941142167 ## Occupancy -0.04819484 0.0101612131 -0.0005422531 -0.0143206543 ## EDAD -0.12960975 -0.0005422531 0.0045323658 -0.0009536548 ## SEXOMASCULINO -2.89411422 -0.0143206543 -0.0009536548 8.5804013484 ``` --- ### Standarized Coefficients - Coefficients depend on the units of measurement of the `\(x\)` - Since `\(x\)` can have different units or magnitudes, we can't directly compare them -- **Example:** `$$\text{ecobici trips}_i=\beta_0+\beta_1\text{temperature}_i+\beta_2\text{polution}_i+u_i$$` -- - It doesn't make sense to compare `\(\beta_1\)` to `\(\beta_2\)` to see what has bigger effect - These variables have very different magnitudes - Increasing temperature by one unit (1 degree celcius) is different than increasing polution by one unit (1 μg/m3) -- - To make them directly comparable, we want to make them unitless (standarized) - Does increasing temperature by .blue[one standard deviation] has the same effect as inreasing polution by .blue[one standard deviation]? --- ### Standarized coeffcients Basically, we standardize all the variables and run the regression: `$$\frac{y_i-\bar{y}}{s_y}=\gamma_1\frac{x_{i1}-\bar{x}_{1}}{s_{x_1}}+\gamma_2\frac{x_{i2}-\bar{x}_2}{s_{x_2}}+...+\gamma_k\frac{x_{ik}-\bar{x}_k}{s_{x_k}}+u_i$$` So then `\(\gamma_k\)` measures the impact of one standard deviation increase of `\(x_k\)` on standard deviation in y -- But there is a short cut to calculate these standard coefficients `$$\gamma_k=\beta_k\frac{s_{x_k}}{s_y}$$` --- ### Example Urgent Care duration example: - `\(s_y=111.82\)` - `\(s_{Age}=20.82\)` - `\(s_{Occupancy}=13.921\)` -- We calculated that `\(\hat{\beta}_{Age}=0.206\)` and `\(\hat{\beta}_{Occupancy}=3.703\)` -- **Standardized coefficients** `$$\begin{align*} & \hat{\gamma}_{Age} =\hat{\beta}_{Age}\frac{s_{Age}}{s_{y}}=0.206\frac{20.82}{111.82}=0.0383 \\ &\hat{\gamma}_{Occupancy} =\hat{\beta}_{Occupancy}\frac{s_{Occupancy}}{s_y}=3.703\frac{13.921}{111.82}=0.461 \end{align*}$$` --- <img src="data:image/png;base64,#Exam_fq.png" width="85%" /> ---